INTRODUCTION
In tribology, the mechanisms impose their constraints and their operating conditions, kinematic, dynamic and thermic to their contacts; the latter, in return can or not support these requests and thus can deteriorate or not the correct operation of these same mechanisms. The interface which is not other than fluid film interposed between surfaces in contact, must support these constraints and ensure relative a correct operation between them. The increasingly severe technological requirements of the mechanisms and engines lead to an increase in dissipated energy; the generated temperature can be very high, then the viscosity is decreases and thus the journal lift, as well as the materials mechanical characteristics. The thermal and mechanical deformations generated, can be considerable and very consequent at such point as they can compromise the working clearance and consequently the minimal film thickness.
After the analysis of the various bibliographies treating the thermoelastohydrodynamic phenomena in the bearings, we present the theory used for the resolution of the problem TEHD.
The effect of elastic deformation of the bearings, on the performance of connecting rod bearings has been studied by many research workers, this shows that is a key factor in the analysis of these bearings., to quote only some, Ferron et al. (1983) note that the effect of dilation of the elements of the journal, compromises the radial clearance under operation.
Boncompain et al. (1986) have shown that the variations found between
the experimentation and theory were due to the thermoelastic effects, Khonsari
and Wang (1991) presented a study including the effect of dilation thermal of
the bearing and/or the crankpin and the elastic strain of the bearing due to
the pressure field; Piffetau et al. (2000) présented numerical
study for the thermoelastohydrodynamic comportment of the rod bearing subject
at a dynamical loading, they show that for the studies cases, the TEHD modeling
does not bring much more precision than isothermal TEHD modeling, like the study
realised one year ago by Garnier et al. (1999) in automotive engine with
four cylinders in line.
Souchet et al. (2001) Presented a thermoelastohydrodynamic study, where they analysed the influences of the conditions boundary. During the same time Hoang and Bonneau (2001) undertake an experimental study for the heating effects on the connecting rod bearings, in the same context and in the same year Byung and Kyung (2001) make a (TEHD) study and show the influences of the heating and mechanical effects on the behavior of a big end journal of the Diesel engine RustonHornsby 6 VEB MkIII, whose study was undertaken before by Oh and Goenka (1985) and thereafter by Kumar et al. (1990).
The taking into account of TEHD analysis. is very recommended in the engines working under severe conditions, this to predict the performances of bearings in internal combustion engines.
Governing equations: The schematic diagram of connecting rod bearing is shown in Fig. 1, both of the value and direction of the load applied to the crankshaft vary with time, so the center of the crankpin has periodic motion relative to the connecting rod bearing center. The X_{0}Y_{0}Z_{0} coordinate system, whose origine is fixed at the undeformed bearing center, is used as a reference coordinate system of the analysis.
The problem is to find the pressure and temperature fields in the lubricant film, in order to find the most important characteristic in fonctionnement, which is the minimum film thickness.
Reynolds equation in transient state: The Reynolds equation is obtained from the NavierStokes equations, in the case of the dynamic mode, the additional data is the variation of the load in module and direction. (Fig. 1).
The crankshaft speeds are written:
For the bearing:
By making the approximation R_{a} R_{c}
R and (C/R) << 1, the Reynolds modified equation is written as:

Fig. 1: 
Schematic diagram 
the dimensionless Reynolds equation is written:
: is the average velocity of the crankpin, reported to the load.
ψ: Load angular velocity
φ: Angular Velocity of the centeres line relatively at load.
ε : Crushing velocity
ω_{a}, ω_{c}: Angular velocities of the crankshaft and bearing.
Mobility method: The second member of the Eq. (2) fact
of appearing t he two unknown factors of the problem: ε and φ. The
traditional solution is to give two values arbitrary to ε and to φ
and to use an iterative method on these two speeds until the hydrodynamic load
W calculated equal and is opposed to the load applied F.
Speeds of the crankshaft center inside the bearing are determined by writing the equality between the hydrodynamic force in film and the whole of the forces applied to the journal:
Relatively for the axis system lied to the centeres line, the Eq.
(3) became:
W_{ε} and W_{φ} are the components of the hydrodynamic load according to the direction of eccentricity and its perpendicular.
Numerical calculations being very significant, we prefer to use the mobility method of Booker (1984) which allows a fast and precise resolution of the problem.
vector of mobility has as components:
α is the angle between the mobility vector and the eccentricity direction.
The Reynolds equation modified is put in the form, while posing :
The second member of the Eq. (7) utilizes only one unknown
factor, the direction of mobility α; the solution of this equation became
simplified. However the determination of α, which for a given eccentricity,
is only function of the angle of chock φ requires to use a numerical method
of interpolation (the relation enters α and φ is a priori unknown).
Moreover the module of the vector mobility M equal to 1 and is reactualized
with the computed value with the step of previous time. The boundary conditions
associated with the Eq. (7) are those of Reynolds, by taking
atmospheric pressure like reference, they are written:
The components of the load without dimension are written:
The module of the load is given by:
What makes it possible to calculate the module of the vector mobility and the angle of chock:
The dimensioned hydrodynamic load is equal to the load applied F and is written:
Since the eccentricity varies in time, therefore it is necessary to choose
an interval of time between two successive points:
et
The minimal film thickness: The minimal film thickness of lubricating, without thermal and elastic deformation, is expressed by:
to which we must add the deformations due to the fields of pressures and the thermal deformations or dilations.
where: δh_{p} is the elastic strain
δh_{d} = δh_{c}  δh_{a} is the differential dilatation between the crankpin and the bearing.
The energy equation: The energy equation permitted to calculate the temperature field in the fluid, it translates the energy conservation and permitted to study the thermal transfers in the journal. According to Boncompain (1984) the energy equation is written:
Where the first term of the first member represents the convection, the first term of the second member conduction and the second term of the second member, viscous dissipation.
Heat conduction equation: Our study takes into account the transfer of heat by conduction in the bearing, In order to determine the thermal deformations of the elements of the journal, the temperatures in the solid elements must be known, the equation of heat (16) is:
Were T_{c} is the temperature in any point of the bearing according to r_{c}, θ and Z
Calculation of final film thickness:
• 
Owing to the fact that the crankpin is massive and in rotation,
owing to the fact that metals are of good thermal drivers relative to other
materials, owing to the fact that the temperature does not have an effect
localised like the pressure, but cumulative until thermal stability, we
estimate that a calculation of simple dilation at an average temperature
of the crankpin is sufficient. 
• 
Contrary to the non revolving bearings, the crank pins are constantly
in rotation, it is not the same surface which is always under constraint,
therefore heat is not localised in the same portion of surface, but distributed
according to its structure. 
We estimate that the ideal and the finality are that after the established
mode and thus thermal stability, the radial clearance under operation is not
compromised and is assured.
Global heating effect: The determination of the distribution of the temperature in lubricating film like in the solids in contact is done by the resolution of the equation of energy, the equation of Reynolds generalized and also the equation of conduction of heat in the crankpin and the bearing. The resolution of this threedimensional problem was considered only very recently.
DISSIPATED POWER
The energy dissipated in the contact is significant and the temperatures in the fluid and contiguous materials with film are raised, it results from it a fall from viscosity and thus a reduction from the bearing pressure of the journal. A simple analysis consists in carrying out a total heat balance and thus determining an average temperature value and viscosity of the lubricant. The average temperature value will be calculated starting from the power dissipated during the cycle, average viscosity being obtained starting from a law of variation of viscosity according to the temperature, generally we neglect the effect of the pressure on viscosity in journal bearings.
The formula of Mac Coull and Walther was retained by the ASTM. It is expressed in the form:
Where υ is the kinematic viscosity in centistoke (mm^{2}/s), T
is the absolute temperature and A, B are the specific constants of the lubricant,
the parameter α depends on viscosity υ.
Where:
represents the power dissipated by load rotation.
represents the power dissipated by the film crushing.
represents the power dissipated by the fluid shear.
Temperature elevation: If it is admitted that the power dissipated in film is evacuated by the lubricant, the rise in the temperature can be written:
With C_{P} specific heat of the lubricant, ρ density of the lubricant at the supply temperature and Q_{moy} the medium flow.
Frêne (1982) and Frêne et al. (1990) propose to approach the maximum value of the temperature by the following empirical relation:
The minimal film thickness will be finally the minimal thickness of the fluid film at which we add displacements due to the pressures (mechanics) and displacements due to the dilatation.
MECHANICAL DEFORMATIONS
The connecting rod material is assumed to be isotropic, the stressstrain relationship can be written as:
Where
Stress analysis for the bearing structure, is accomplished by the finite elements software CASTEM 2000, with a pressure of 1Mpa, the meshing used is hexaedric of 8 nodes.
Where M_{f} (θ, z)is the compliance matrix, obtained for a unit
pressure of 1Mpa, displacement in each point is obtained by multiplication of
the coefficient of the matrix by the corresponding pressure of the node considered,
in this way we do not have recourse to each iteration on the pressure to calculate
by finite elements displacement (Bouyer and Fillon, 2003).
The calculation of the mechanical deformations is applied only to the bearing, the crankpin relative with the bearing is bulkier and the deformations are negligible.
Dilatation effect: Both of the crankpin and the bearing are dilated.
For the crankpin and the bearing:
R_{a}, R_{c} are respectively the crankpin and the bearing radius.
T_{a}, T_{c} are respectively the crankpin and bearing average temperatures.
The two effects combined bring to a differential expansion and must be added to the film thickness.
Resolution procedure: To determine the pressure field in lubricating
film, finite difference method is applied for the resolution of the modified
Reynolds Eq. (8). The associated linear system cannot be solved
directly, because of use of the boundary condition of Reynolds (relations (9),
we thus applied the iterative method of Gauss Seidel with surrelaxation coefficient,
the calculation of deformations was done according to laws' of elasticity and
is solved by the known software CASTEM 2000.
Displacements are given for a unit pressure of 1Mpa, to have the real displacement of a node it is necessary to multiply the matrix compliance obtained, by the pressure in this node (Byung and Kyung, 2001), (Bouyer and Fillon, 2003).
The reference pressure is the atmospheric pressure, the temperature is determinate by the heat flow in the journal, due to heat transfer in the bearing and the viscous dissipation in lubricant. The temperatures in film and bearing interface are equal (thin film), the thermal transfer in the crankpin is neglected And we consider for this last, only dilatation between an average and reference temperatures.
the pressures in the sections of entry and of exit of film are equal to the supply pressure, that of the edges to the atmospheric pressure.
For displacements, the bearing being embedded in the big end, radial displacements on the level of the external radius of the bearing are taken as null.

Fig. 3: 
Distribution pressure for some crankshaftangles 
Simulation results: We have made our calculation in the connecting rod
bearing of Ruston and Hornsby 6vebx MkIII4 Stroke Diesel engine, as the bearing
has a full circumferential oil groove, the calculation is performed on a single
land of the bearing.
Table 1: 
Journal bearing data and operating conditions 


Fig. 4: 
Minimum film thickness 

Fig. 5: 
Maximum film pressure 
The load diagram at 600 rpm is in Fig. 2. Bearing dimensions,
material properties and operating conditions are listed in Table
1. Boundary conditions are Reynolds conditions. Figure
3 shows the three dimensional pressure distribution for some crankshaft
angles, the minimal film thickness, the maximal film pressure with and without
deformation effect, the dissipater power, outlet flow rate and the rubbing torque
are represented respectively in Fig. 48.
We remarked that to take the thermoelastothermodynamic analysis in consideration,
shows that all the bearing characteristics vary and among all these characteristics
the minimal film thickness is the most important and must be assured, because
it characterizes the working clearance.
CONCLUSION
A thermoelastohydrodynamic lubrication analysis of connecting rod bearing is proposed which includes thermal dilatation and elastic deformation of the bearing surface. Simulation results show that the thermal distortion has remarkable effects on the bearing performance such as the minimum film thickness, maximum film pressure and oil flow rate. It is concluded that the thermoelastohydrodynamic lubrication analysis is very recommended to predict the performance of connecting rod bearings in internal combustion engines.
Our results within sight of the results obtained by other researchers are concordant, the few differences obtained are due to the boundary conditions and the methods of resolutions.
Appendix:
Thermal deformation of the bearing
According to Laroze (1991), the equation of heat (16) can be written,
By integration, we obtain:
r_{1} and r_{2} are respectively the internal and external
radius, T_{1} and T_{2} are respectively the temperatures of
the internal and external walls of the bearing.
By considering the assumption of plane deformation and according to the equation of Navier we have:
Where u(r) represents radial displacement. By integration successive, we have:
With:
For the crankpin:
Average
temperature of the crankpin
T_{0}Temperature of reference to which dimensions were taken
NOMENCLATURE
C 
: 
Radial clearance (RcRa) 
Ca, C_{c} 
: 
Friction torques on the crankpin and bearing 
CP 
: 
Specific heat of the lubricant 
D 
: 
Average diameter of the journal 
e 
: 
Eccentricity of the journal 
F 
: 
Dynamic load applied to the journal 
h 
: 
Film thickness H: Dimensinless film thickness (h/C) 
h_{min} 
: 
Minimal film thickness 
L 
: 
Length of the bearing 
L2 
: 
Crank radius 
L3 
: 
Rod length 
M 
: 
Mobility factor 
m 
: 
Rod mass 
M_{ε_}, M_{φ_} 
: 
Components of the mobility vector 
Oa 
: 
Crankpin Center 
Oc 
: 
Bearing Center 
O_{c}X_{0}Y_{0} 
: 
Axis System fixed to the rod 
O XYZ 
: 
Axis System related to the centers locus 
P 
: 
Hydrodynamic pressure in film 

: 
Dimensionless Pressure . 

: 
Dissipated power 
Q 
: 
Axial flow 

: 
Dimensionless Axial flow. 
R 
: 
Journal average radius 
R_{a} 
: 
Crankpin radius. 
R_{c} 
: 
Bearing radius. 
t 
: 
Time 
T 
: 
Temperature 
T_{a} 
: 
Supply temperature 
W 
: 
Hydrodynamic load supported by the journal 
x 
: 
Coordinate in the (circumferential direction) 
y 
: 
Coordinate in the film thickness direction. 
z 
: 
Coordinate in the axial direction 
α 
: 
Mobility direction. 
Δ τ 
: 
Time step. 
Δ T 
: 
Temperature rise 
ε 
: 
Relative eccentricity 
θ 
: 
Circumferential coordinate in the Oxyz system 
θ_{2} 
: 
Swing angle of the crankshaft 
θ_{E}, θ_{S} 
: 
Circumferential coordinates of the entry and the exit of lubricating
film 
μ 
: 
Lubricant dynamic viscosity. 
ρ 
: 
Lubricant density. 
ι 
: 
Angle between the load F and axis OX. 
ω 
: 
Average angular velocity of the crankshaft and the bearing. 
ω_{α}, ω_{C} 
: 
Angular velocities of the crankshaft and bearing brought back to O_{c
}X_{0}Y_{0 } 
ψ 
: 
Load angular velocity 
φ 
: 
Angular velocity of calage angle. 